301 Moved Permanently - Free Printable

Let's solve each of the linear equations in one variable step by step. I'll go through them systematically and explain the process.

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(1) $ 8x = 6x + 10 $

Step 1: Subtract $6x$ from both sides
$ 8x - 6x = 10 $
$ 2x = 10 $

Step 2: Divide both sides by 2
$ x = \frac{10}{2} = 5 $

✔ Answer: $ x = 5 $

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(2) $ 4 = 5x - 6 $

Step 1: Add 6 to both sides
$ 4 + 6 = 5x $
$ 10 = 5x $

Step 2: Divide by 5
$ x = \frac{10}{5} = 2 $

✔ Answer: $ x = 2 $

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(3) $ 13y = -12y + 100 $

Step 1: Add $12y$ to both sides
$ 13y + 12y = 100 $
$ 25y = 100 $

Step 2: Divide by 25
$ y = \frac{100}{25} = 4 $

✔ Answer: $ y = 4 $

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(4) $ 18x = -13x + 62 $

Step 1: Add $13x$ to both sides
$ 18x + 13x = 62 $
$ 31x = 62 $

Step 2: Divide by 31
$ x = \frac{62}{31} = 2 $

✔ Answer: $ x = 2 $

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(5) $ 5x + (-3) = 12 $ → $ 5x - 3 = 12 $

Step 1: Add 3 to both sides
$ 5x = 15 $

Step 2: Divide by 5
$ x = 3 $

✔ Answer: $ x = 3 $

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(6) $ 3(x + 1) = 6 $

Step 1: Distribute
$ 3x + 3 = 6 $

Step 2: Subtract 3
$ 3x = 3 $

Step 3: Divide by 3
$ x = 1 $

✔ Answer: $ x = 1 $

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(7) $ 7(m - 9) = 35 $

Step 1: Divide both sides by 7
$ m - 9 = 5 $

Step 2: Add 9
$ m = 14 $

✔ Answer: $ m = 14 $

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(8) $ 8(x + 3) + 2 = 42 $

Step 1: Expand
$ 8x + 24 + 2 = 42 $
$ 8x + 26 = 42 $

Step 2: Subtract 26
$ 8x = 16 $

Step 3: Divide by 8
$ x = 2 $

✔ Answer: $ x = 2 $

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(9) $ 16 - 3(x - 7) = -14 $

Step 1: Expand
$ 16 - 3x + 21 = -14 $
$ -3x + 37 = -14 $

Step 2: Subtract 37
$ -3x = -51 $

Step 3: Divide by -3
$ x = 17 $

✔ Answer: $ x = 17 $

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(10) $ 3(x + 5) = 15 $

Step 1: Divide both sides by 3
$ x + 5 = 5 $

Step 2: Subtract 5
$ x = 0 $

✔ Answer: $ x = 0 $

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(11) $ 12(3 - x) = 48 $

Step 1: Divide both sides by 12
$ 3 - x = 4 $

Step 2: Subtract 3
$ -x = 1 $

Step 3: Multiply by -1
$ x = -1 $

✔ Answer: $ x = -1 $

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(12) $ 5x + 8(2x - 9) = 54 $

Step 1: Expand
$ 5x + 16x - 72 = 54 $
$ 21x - 72 = 54 $

Step 2: Add 72
$ 21x = 126 $

Step 3: Divide by 21
$ x = 6 $

✔ Answer: $ x = 6 $

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(13) $ \frac{x}{6} = 5 $

Multiply both sides by 6:
$ x = 30 $

✔ Answer: $ x = 30 $

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(14) $ \frac{m}{3} = 4 $

Multiply both sides by 3:
$ m = 12 $

✔ Answer: $ m = 12 $

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(15) $ \frac{m}{4} = \frac{1}{2} $

Multiply both sides by 4:
$ m = 2 $

✔ Answer: $ m = 2 $

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(16) $ \frac{x}{11} = 6 $

Multiply both sides by 11:
$ x = 66 $

✔ Answer: $ x = 66 $

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(17) $ \frac{x}{5} = 7 $

Multiply both sides by 5:
$ x = 35 $

✔ Answer: $ x = 35 $

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(18) $ \frac{x}{7} = 4.5 $

Multiply both sides by 7:
$ x = 4.5 \times 7 = 31.5 $

✔ Answer: $ x = 31.5 $ or $ \frac{63}{2} $

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(19) $ \frac{r}{9} = -11 $

Multiply both sides by 9:
$ r = -99 $

✔ Answer: $ r = -99 $

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(20) $ \frac{x}{-4} = \frac{1}{8} $

Multiply both sides by -4:
$ x = \frac{1}{8} \times (-4) = -\frac{4}{8} = -\frac{1}{2} $

✔ Answer: $ x = -\frac{1}{2} $

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(21) $ \frac{x}{-4} = \frac{3}{4} $

Multiply both sides by -4:
$ x = \frac{3}{4} \times (-4) = -3 $

✔ Answer: $ x = -3 $

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(22) $ \frac{x}{2} = -39 $

Multiply both sides by 2:
$ x = -78 $

✔ Answer: $ x = -78 $

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(23) $ \frac{5z - 7}{3z} = 2 $

Step 1: Multiply both sides by $3z$:
$ 5z - 7 = 2 \cdot 3z = 6z $

Step 2: Subtract $5z$:
$ -7 = z $

✔ Answer: $ z = -7 $

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(24) $ \frac{2y + 5}{y + 4} = 1 $

Step 1: Multiply both sides by $y + 4$:
$ 2y + 5 = y + 4 $

Step 2: Subtract $y$:
$ y + 5 = 4 $

Step 3: Subtract 5:
$ y = -1 $

✔ Answer: $ y = -1 $

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(25) $ \frac{2x + 1}{3x - 2} = \frac{5}{9} $

Step 1: Cross-multiply:
$ 9(2x + 1) = 5(3x - 2) $

Step 2: Expand:
$ 18x + 9 = 15x - 10 $

Step 3: Subtract $15x$:
$ 3x + 9 = -10 $

Step 4: Subtract 9:
$ 3x = -19 $

Step 5: Divide by 3:
$ x = -\frac{19}{3} $

✔ Answer: $ x = -\frac{19}{3} $

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(26) $ \frac{2x}{3x + 1} = -3 $

Step 1: Multiply both sides by $3x + 1$:
$ 2x = -3(3x + 1) $

Step 2: Expand:
$ 2x = -9x - 3 $

Step 3: Add $9x$:
$ 11x = -3 $

Step 4: Divide by 11:
$ x = -\frac{3}{11} $

✔ Answer: $ x = -\frac{3}{11} $

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(27) $ \frac{5y}{3} + \frac{2}{5} = 1 $

Step 1: Subtract $\frac{2}{5}$:
$ \frac{5y}{3} = 1 - \frac{2}{5} = \frac{3}{5} $

Step 2: Multiply both sides by 3:
$ 5y = \frac{9}{5} $

Step 3: Divide by 5:
$ y = \frac{9}{25} $

✔ Answer: $ y = \frac{9}{25} $

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(28) $ \frac{x}{2} + \frac{x}{3} + \frac{x}{4} = 13 $

Step 1: Find LCM of denominators (LCM of 2,3,4 = 12)

Convert each term:
$ \frac{6x}{12} + \frac{4x}{12} + \frac{3x}{12} = 13 $
$ \frac{13x}{12} = 13 $

Step 2: Multiply both sides by 12:
$ 13x = 156 $

Step 3: Divide by 13:
$ x = 12 $

✔ Answer: $ x = 12 $

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(29) $ \frac{2x}{3} - \frac{3x}{8} = \frac{7}{12} $

Step 1: Find LCM of denominators: LCM of 3, 8, 12 = 24

Convert:
$ \frac{16x}{24} - \frac{9x}{24} = \frac{7}{12} $
$ \frac{7x}{24} = \frac{7}{12} $

Step 2: Multiply both sides by 24:
$ 7x = 14 $

Step 3: Divide by 7:
$ x = 2 $

✔ Answer: $ x = 2 $

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| (1) | $ x = 5 $ |
| (2) | $ x = 2 $ |
| (3) | $ y = 4 $ |
| (4) | $ x = 2 $ |
| (5) | $ x = 3 $ |
| (6) | $ x = 1 $ |
| (7) | $ m = 14 $ |
| (8) | $ x = 2 $ |
| (9) | $ x = 17 $ |
| (10) | $ x = 0 $ |
| (11) | $ x = -1 $ |
| (12) | $ x = 6 $ |
| (13) | $ x = 30 $ |
| (14) | $ m = 12 $ |
| (15) | $ m = 2 $ |
| (16) | $ x = 66 $ |
| (17) | $ x = 35 $ |
| (18) | $ x = 31.5 $ |
| (19) | $ r = -99 $ |
| (20) | $ x = -\frac{1}{2} $ |
| (21) | $ x = -3 $ |
| (22) | $ x = -78 $ |
| (23) | $ z = -7 $ |
| (24) | $ y = -1 $ |
| (25) | $ x = -\frac{19}{3} $ |
| (26) | $ x = -\frac{3}{11} $ |
| (27) | $ y = \frac{9}{25} $ |
| (28) | $ x = 12 $ |
| (29) | $ x = 2 $ |

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